We investigate spaces of spaces in the category of represented spaces and type-2 computable maps. Concretely, such spaces of spaces are given by bundles whose bases are names for spaces. We give natural examples of such bundles for Polish and compact Polish spaces and show they are essentially equivalent. We then propose definitions of genericity and computable categoricity, according to which the Cantor space is computably categorical and generic as a compact Polish space. We also show that the degree of categoricity of \(\mathcal {S}^1\) is the Weihrauch degree \(\lim \) .